4 votes
Accepted

Royal road to Abstract Harmonic Analysis?

Rudin's 'Fourier analysis on groups' is one of the most enjoyable books I ever read. How well versed are you in classical harmonic analysis? the books by Katznelson and by Helson go a long way toward ...
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  • 12.9k
3 votes
Accepted

Cohomology over "discrete reals" vs "continuous reals"?

Because $\mathbb{R}$ is contractible, so are all $B^n\mathbb{R}_{\text{continuous}}$. So $H^n(X;\mathbb{R}_{\text{continuous}}) = 0$ for all $X$.
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  • 150
2 votes

Reference request: Tiling the plane with congruent snowflakes

I couldn't find any discussion of this problem online, so I went ahead myself and made an algorithm which can exhaustively generate the figures described, given a few extra assumptions. The linked ...
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2 votes
Accepted

How many connected bipartite graphs are there?

The probability that the uniform random graph will be bipartite is in fact extremely low. It is not straightforward to compute exactly, because there is a large number of ways for it to happen, and ...
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2 votes

Herbert Amann Analysis vs Zorich Analysis

I was interested in this question, too, but also nowhere found a direct answer to it, except on a german website called matheplanet (you may translate the reviews/comments). But I can tell you ...
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  • 21
2 votes
Accepted

Generalized Stochastic Matrices with Negative Transition Probabilities

You need more than $\ |\lambda_j|\le1\ $ to get convergence of $\ A^k\ $. If $$ A=\pmatrix{1&0&0\\0&-1&2\\ 0&0&1}\ , $$ for instance, the eigenvalues of $\ A\ $ are $\ 1,-1\ $ ...
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2 votes

First-order logic with finitistic approach

Here is as paper on a formal system for "feasible" numbers. As Alex Kruckman suggests, the paper often just refers to existing theory on formal logic for background. The main novel ...
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  • 2,659
2 votes
Accepted

Attempting to restate the question of whether the collatz conjecture has a nontrivial cycle as a combinatorics problem

Just to add some more combinatorical information. Preamble: I'm used to use the following letters, which are different from yours, and I'm lazy to adapt this, please bare with me... I use $N$ for ...
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2 votes

Higher-Order Differential Operators as Vector Fields

The velocity of a particle on a curved space is a covariant notion but acceleration isn't. They appear to be very similar concepts in flat space because we represent them in the same way, but on ...
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1 vote
Accepted

Defining a Coxeter group using all reflections

What you're asking about is what's sometimes called the "dual approach" to reflection groups and their braid groups. A good place to start would be D. Bessis, The Dual Braid Monoid, https:/...
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  • 71
1 vote
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Reference request for elementary geometry.

I can think on several options, one of them is Elementary Geometry from an advanced standpoint by E.Moise. But if you are looking for exercises like in math olympiads, there is a lot of material like ...
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1 vote
Accepted

Radical axis generalization

Since this is in part a reference request, I'd recommend the paper Pfiefer and van Hook, Circles, Vectors, and Linear Algebra. (In the following I'm going to abuse notation and switch freely between $\...
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  • 6,266
1 vote
Accepted

Reference request for Fenchel-Rockafellar duality for dual system

Actually for a dual system it suffices $X$ and $Y$ to be linear spaces, i.e., no apriori topological structure.They get their topological structures from the weak topologies. In the presence of (...
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1 vote

Which branch of math studies this problem?

Technically, you haven't reduced the number of computations, as you first needed to calculate $5 \times 10^6 + 7$ before squaring it, and must also find a way to extract the squares from the new ...
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  • 4,431
1 vote

Striking applications of linearity of expectation

It also helps calculate the falling factorial momentum of the Poisson distribution. As recall: The falling factorial momentum is defined as: $$\mathbb E[(X)_r]=\mathbb E[X(X-1)\cdots(X-r+1)]=r!\...
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  • 5,281
1 vote

Symmetric part of the inverse matrix

In case $\operatorname{Sym}(A)$ is positive definite, we have $\operatorname{Sym}(A^{-1})\preceq\operatorname{Sym}(A)^{-1}$. Let $A=P(I-K)P$ where $P=\operatorname{Sym}(A)^{1/2}\succ0$ and $K$ is skew-...
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  • 123k

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